Mathematical Sciences: Geometric Topology and the Fundamental Group

几何群是在标准几何上适当地、不连续地和上紧地作用的群。这位首席研究者试图理解几何群的组合学：几何学对组合学的含义是什么？几何群的特征是什么组合性质？这位研究人员刚刚建立了3维上紧和有限体积双曲群的组合特征。他还在创建计算机程序，以帮助研究与双曲群和欧几里德群相关的递归计算结构。研究人员计划的下一步是：(1)以尽可能多的不同方式应用他的刻画(这是否意味着3维的负曲群可以实现为共形群？瑟斯顿的双曲化定理能从刻画中推导出来吗？(2)用计算机程序研究由生成元和关系子给出的有趣的无限群类(例如，Coxeter群G(3，7，n)和Conway的Fibonacci群)。这个项目的工作利用几何学和群论之间微妙的相互作用来推进每个领域的状态。

A geometric group is a group acting properly discontinuously and cocompactly on a standard geometry. This principal investigator is attempting to understand the combinatorics of geometric groups: What are the implications of the geometry for the combinatorics? What combinatorial properties characterize a geometric group? The investigator has just established a combinatorial characterization of cocompact and finite-volume hyperbolic groups in dimension 3. He is also in the midst of the creation of computer programs to aid in studying the recursive computational structures associated with hyperbolic and Euclidean groups. The next steps in the investigator's plan are the following: (1) apply his characterization in as many different ways as possible (Does it imply that negatively curved groups in dimensions 3 can be realized as conformal groups? Can Thurston's hyperbolization theorem be deduced from the characterization? Can space-filling curves be so studied?); (2) use the computer programs to study interesting infinite classes of groups given by generators and relators (e.g., Coxeter's groups G(3,7,n) and Conway's Fibonacci groups). The work of this project uses the subtle interplay between geometry and group theory to advance the state of each field.